DRAM: A Model of Health Care Resource Allocation

DRAM: A MODEL OF HEALTH CARE RESOURCE ALLOCATION

David Hughes Andrzej Wierzbicki

RR-80-23 May 1980

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS Laxenburg, Austria

Research Reports, which record research conducted at IIASA, are independently reviewed before publication. However, the views and opinions they express are not necessarily those of the Institute or the National Member Organizations that support it.

Copyright O 1980

International Institute for Applied Systems Analysis

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher.

FOREWORD

The principal aim of health care research at IIASA has been to develop a family of submodels of national health care systems for use by health service planners.

The modeling work is proceeding along the lines proposed in the Institute's current Research Plan. It involves the construction of linked submodels dealing with population, disease prevalence, resource need, resource allocation, and resource supply.

This is the second research report o n the disaggregated resource allocation sub-model called DRAM. It describes the extension of the Mark 1 version (RR-78-8) t o include the distribution of many resources across different modes of care. The earlier assumption that all available resources must be used has been relaxed, and an extensive analytic treatment suggests various methods for estimating the submodel's parameters. Several case studies that use the model are in progress and reports on these applications will be forthcoming.

This paper is an output of a collaboration between two Areas at IIASA. It describes how a health resource allocation model, developed in the Health Care Systems Task of the Human Settlements and Services Area, may be solved by using optimization techniques studied in the Optimization Task of the System and Decision Sciences Area.

Related publications in Health Care Systems and in Optimization are listed at the end of this report.

A N D R E I ROGERS Chairman

Human Settlements and Services Area

A N D R Z E J WIERZBICKI Chairman

System and Decision Sciences Area

ACKNOWLEDGMENTS

We are indebted to many colleagues at llASA and elsewhere for their help and advice. We would like to offer special thanks to Rebecca Crow, who prepared the typescript, tables, and diagrams, and t o the staff of the South Western Regional Health Authority, who provided the data used in Section 5.

CONTENTS

1 INTRODUCTION

2 MODEL FORMULATION AND SOLUTION 2.1 Notation and Assumptions

2.2 The Simple Model 2.3 Three Extensions 2.4 Solution Procedure

3 SOLUTION CHARACTERISTICS 3.1 The Simplest DRAM

3.2 Conditions for Fitting X, Y to Two Data Points 3 3 Conditions for Fitting a,

0

to Two Data Points 3.4 Conditions for Fitting X,

Y

or a,

0

to One Data Point 3.5 Conditions for Fitting X,

Y,

a,

0

to Two Data Points 4 ESTIMATION OF PARAMETERS

4.1 Parameters and Data 4.2 Estimation of X,

Y

4.3 Estimation of a,

0

4.4 Estimation of a,

0

and

X, Y

4.5 An Alternative Approach 4.6 Estimation of C

5 EXAMPLES

5.1 Application of DRAM 5.2 Example 1 : Hospital Beds

5.3 Example 2: The Balance of Inpatient and Outpatient Care 6 SUMMARY

REFERENCES APPENDIXES

A An Alternative Formulation o f DRAM

B

Computer Programs and Solution Efficiency C Fitting Four Parameters t o Four Data Points D Unbiased Regression Estimators

E Sensitivity o f the Solution to Parameter Changes F List o f Principal Symbols

viii

1 INTRODUCTION

It has been widely observed (Feldstein 1967, Van der Gaag et al. 1975, Rousseau 1977) that the demand for health care seems t o be insatiable. When more hospitals are opened, more patients are treated, and the hope expressed at the inception of the U.K. National Health Service that increasing supplies of health care would reduce subsequent demands has not been realized there or anywhere. The causes of this phenomenon are various, but it gives rise t o the same question in all countries: What health care resources should be made avail- able?

Unfortunately, the principal output of health care systems - health - is almost impossible t o define or measure (Cardus and Thrall 1977). Much as we would like t o design a health care system that would maximize health, we do not even know how t o begin. Instead, we seek to predict how those hos- pitals and other resources available in the health care system (HCS) will be used.

Who gets what?

DRAM (a disaggregated resource allocation model) is designed t o help answer such questions. It is one of the submodels of the HCS model conceived by Venedictov e t al. (1977), and being developed by a group of scientists from different countries working at the International Institute for Applied Systems Analysis (IIASA). Figure 1 shows the five groups of submodels of the HCS devel- oped so far at IIASA; they are explained in more detail in a recent status report (Shigan e t al. 1979). This figure represents one part of the complete HCS: the processes by which people fall ill and by which resources are provided and used for their care. DRAM (in the group of resource allocation submodels) represents how the HCS allocates limited supplies of resources among competing demands of morbidity. Specifically, it asks If a certain mix of health care resources (e.g., hospital beds, nursing care) is available, how will the HCS distribute them among patients? DRAM does not prescribe an optimal allocation of resources.

Resource

FIGURE 1 The family of HCS submodels constructed at IIASA.

Instead, it simulates how the HCS responds when resource availability changes.

Even in countries with market economies, there are invariably some planning instruments for controlling the supply of public goods. But even in countries with planned economies, resources cannot be allocated in a rigid, centralized manner. In every country, doctors have clinical responsibility for their patients, and the pattern of care is determined by many local decisions. McDonald et al.

(1974), Rousseau (1977), and Burton et al. (1978) are among those who have modeled this behavior, and DRAM has close links with the first of these models.

Other models for health care resource allocation were reviewed by Gibbs (1 977) and Nackel et al. (1 978).

Like many models, DRAM has accounting and behavioral components. In the accounting in DRAM, different types of resources are distributed among patients

In different categories (e.g., age, diagnosis)

In different modes of care (e.g., inpatient, outpatient)

a With different levels of resources per patient (e.g., length of stay in hospital)

and no more resources are allocated than are available. The resources can be determined by a resource supply/production submodel such as the IIASA sub- model described in Shigan et al. (1 979), or they can be set by the user as a trial policy option.

The behavioral assumption in DRAM is that the HCS behaves as if it were maximizing a preference function that increases with the number of patients treated and the resources received by each. Some of the parameters in this function represent demand inputs, like the ideal levels at which patients would be treated and would receive resources if no constraints on resource availability existed. These parameters indicate the true "needs" for health care. Other parameters represent the elasticities of the actual levels t o changes in resource

supply, and the balance between need and supply. The relative costs of differ- ent resources are other parameters used by DRAM t o choose between alternative resource mixes. DRAM does not try t o include explicitly every behavioral influ- ence that could be active, but t o use parameters that can represent the results of all these influences. Because the parameters have meanings outside the model, they can be estimated by methods that d o not involve the assumptions underlying DRAM.

Gibbs (1978b) formulated a pilot Mark 1 version of DRAM. This report is the successor, and summarizes progress up t o April 1979. Some but not all of the results have appeared in interim IIASA papers, a list of which appears at the end of this report. Much of this report is about the mathematics of the model, and the examples are concerned with hospital services. Our interests, however, are not so restricted. DRAM is designed t o model the concept that the HCS balances the desirabilities of more individuals receiving care against higher aver- age levels of care. Such a model should be applicable in many sectors of health care, and perhaps also in other public sectors.

Readers who are uninterested in mathematical details can skip to Section 5 t o read about the use of mathematical models in general and t o see examples of DRAM. Two examples are presented: one investigates how hospital beds might be used by the HCS, and the other how the balance between inpatient and outpatient care might change. The other parts of the report develop mathe- matical results that are needed t o support such applications. Section 2 solves the simple DRAM and gives three extensions in which certain restrictions applied t o the simple model are removed. Not every resource allocation pattern can be simulated by DRAM, so Section 3 investigates its admissible solutions. This is a way of explaining the implications of DRAM'S underlying hypothesis. Section 4 presents methods for calibrating DRAM so that it is appropriate for different questions of policy in different regions. The associated computer programs are not described in this report, but Appendix B provides brief details. Section 6 gives a concise summary of the whole report.

2 MODEL FORMULATION AND SOLUTION

The first step in formulating DRAM is t o define variables and t o make the key assumptions in the model precise. This is done in Section 2.1, and Section 2.2 analyzes a simple version of DRAM in which all the available resources must be used. Three extensions of the model are analyzed in Section 2.3, and Section 2.4 describes a computational method that can be used t o solve all four versions of DRAM.

2.1 Notation and Assumptions

We use the indices j = patient category ( j = 1, 2,

. . . ,

J), k = mode of care (k = 1, 2,

. . .

^, K), and 1 = resource type (1 = 1, 2,

. . .

, L ) in defining the model variables

xjk = numbers of individuals in patient category j who receive resources in mode of care k (per head of population per year)

yjkl = supply of resource type 1 received by each individual in patient category j in mode of care k

and in writing Cj Ck xjkyjkl as the total resources of type I that are allocated (per head of population per year). DRAM seeks to determine xjk, yjkl V j , k, I , within constraints on total resources, so as t o maximize a function

C, X , Y, a , /3 are model parameters ( C denotes {C,, I ⁼ 1, 2,

. . . ,

L) and so on).

The monotonically increasing, concave power functions (2) and (3) follow from general assumptions about aggregate behavior in the HCS. They depict the many agents who control the allocation of health care resources as seeking t o attain ideal levels of service ( X ) and supply (Y), but where the urge t o increase the actual levels of service (x) and supply ( y ) decreases with increasing values of x and y , according t o the parameters a and

0.

The costs of different resources (C) are introduced so that marginal increases in U when ideal levels are achieved

( X = X, y = Y ) equal the marginal resource costs. This interpretation is a useful way of introducing meaningful parameters into the model, and Section 4 sug- gests various ways of estimating X, Y, a ,

0,

C in different applications. For the moment, however, we assume these parameters t o be known.

Alternative forms for U(x, y ) can be suggested, and some were analyzed by Hughes (1978b). Appendix A presents one of these and shows that minor changes can greatly change both the characteristics of model predictions and the ease of solution. Equations (1 k ( 3 ) have convenient analytic properties that make it easy to solve this formulation of the model.

2.2 The Simple Model

We seek a solution for x , y that maximizes Eq. (1) subject t o the constraints 0

<

xjk

<

Xjk 0

<

^yjkl

<

^{YjkI (4)}

In this section, we assume that all available resources of type I , R,, must be used.

With Lagrange multipliers A,,

VI,

we adjoin t h e equality constraint, Eq. (5), t o the function that is t o be maximized, Eq. ( I ) , t o give

When certain convexity and concavity assumptions are satisfied (proved below), t h e values of x , y , h that solve the primal problem of rnax,,, mink H ( x , y , A) also solve the dual problem of mink rnax,,, H(x, y , A). T h e optimal values i , j are readily found t o be

where pjk is a weighted sum

7

~1 yjklvjkl

pjk =

7

^{~1 yjkl (9)}

of t h e terms

Vjkl [(Pjkl

+

l ) ( ~ l / ~ l ) P i k l / ( p ~ k l + I ) - 1

1

^/Pjkl (10) and substituting these values into Eq. (6) yields

However, these solutions for x , y are not determined until we find a value

X

that minimizes ~ ( h ) .

In order t o see whether this is possible, we inspect the gradient vector of first derivatives evaluated at x = i(X), y = j ( h ) . After much simplification, this is simply t h e vector with elements

~ H ( A )

- -

ax,

^{- Fl[i(X),}

The corresponding Hessian matrix of second derivatives

Hkk

can be written as t h e sum of t w o matrices

Xik Yikl l / ( f l J % l + l ) a ~ j k ( h )

[ P . (h)]-(aj+2)/("j+l) -

1k

axrn

where

a ~ j k ( h ) ^{- -} y r n ( h ) - l i ( & k r n + I )

axrn

C c r n y j k r n rn

ern

and where the Kronecker delta

a,,

is 1 when 1 equals m , and 0 otherwise. A is a diagonal matrix with all elements positive. Therefore, any quadratic form z f A z is always positive, as are all t h e eigenvalues o f A . Equivalently,A is positive definite. T h e matrix B is symmetric, with typical quadratic forms

z f B z =

C

b l m z 1 z m

lrn

which are non-negative. Therefore, B is positive semidefinite. It follows that

&,

is symmetric and positive definite, and this guarantees that H ( X ) is strongly convex. Finally, it can be shown that H ( X ) therefore has a unique minimum for some h

= X .

In order t o prove that this solution t o t h e dual problem also solves the primal problem, we consider t h e matrix of second derivatives of H ( x , y , A ) with respect t o t h e primal variables z = ( x , y ) , evaluated a t x = i ( X ) , y ⁼ ? ( A ) . In

not only t h e off-diagonal submatrices, b u t all t h e off-diagonal terms are zero.

The remaining diagonal elements

are negative, so that H , , [ i ( X ) , ? ( A ) , A] is negative definite. This is sufficient t o ensure that the solution [x(X), j ( X ) ] is the saddle point t o H ( x , y , A), and thus solves b o t h dual and primal problems.

It remains t o consider the range of possible solutions for A. As any XI tends t o zero, all the elements of

&

tend to minus infmity. We deduce therefore that

XI >

0 for all 1. In order for the solutions (7) and (8) t o satisfy the con- straints (4), we should have

XI >

C, for all I. Unfortunately, this cannot be guaranteed, and examples can be found that use all the resources but exceed the ideal standards X, Y. These unrealistic solutions are a deficiency of this simple formulation of DRAM which can be overcome by extending the model.

2.3 Three Extensions

In the first extension of the simple model, we remove the constraint on indi- vidual resource types (5) and add a constraint on total finance. We seek a solution for x, y that maximizes Eq. (1) subject t o constraints (4) and

This solution is the optimal allocation under the assumption that finance M should be used t o purchase resources that will maximize the returns of Eq. (1).

This assumption is not so realistic for our applications, but it gives a model that is easy t o solve.

We find that the optimal values i ,

p

are the same as solutions (7) and (8), but that the Lagrange multiplier X is now constant across all resource types, A , = A, =

. . .

= A,. The dual function H(X) is a function of a single Lagrange multiplier, X1 say, and using the earlier results, we can show that it is the sum of a set of strongly convex functions. It is therefore also strongly convex with a unique minimum for some value

X1 >

0.

In fact, we can demonstrate a stronger result for this version of the model.

Because

aH(x)

A, ⁼ 1 - = M

- C C C C ~ X ~ ~ Y , ~ ~ < o

a x l i k l

and

>

0 1 < A l

<

^w

ax:

we deduce that there is a unique optimal value

h1 >

1 that minimizes H(x), provided only that the finance available is less than that required t o satisfy all demands M

<

Zj Zk Cl CIXjkYjkl. In other words, there is always a unique resource mix that will maximize perceived preferences.

In the second extension of the simple model, we replace the equality resource constraint (5) by an inequality constraint

F , ( x , y ) - r l = O ; r , > O b'l (15)

where rl represents the unused resources of type I, which must always be non- negative. It is easy t o show that there always exists a point ( x , y , r ) that satis- fies constraints (4) and (1 5 ) provided that the inequality

is satisfied. When sufficient resources of some type are available t o violate Eq. (1 6 ) , it means that there are more than enough of these resources, and that there is no allocation problem! The resource type in excess can be removed from the model.

It is also possible t o show that the model can have no solutions with ijk = 0, j j k l = 0 , o r ijk = X j k . In other words, these constraints are never active. This is because the first two conditions imply that U(x, y ) = - OO,

and because the last condition requires A, ⁼ 1, VI, which causes constraint (1 5 ) t o contradict ( 1 6 ) . We conclude then that the only constraints that can be active are the upper constraint on y and the lower constraint on r .

There are now just two possibilities. The first possibility is that

Yjkl <

Yjkr for all 1. Inspection of the function

shows that it is maximized when r, is zero for all I. The problem is then identi- cal to that analyzed above, and all the previous results hold true. The second possibility is that ,Gjkl = Yjk, for one or more (but not L ) resource types I. From Eq. ( 8 ) , the associated values of ( h l / C , ) are unity, and the rest of the problem is equivalent t o the dual problem specified in Section 2.2, but with the extra con- straint

The third extension of the simple model subtracts the costs of the used resources from the preference function

~ ( x , Y ) =

1

^{Cgjk(xjk) +}

CC C X . ~ ~ ~ ~ ~ ~ ( Y ~ ~ ~ )

i k j k l

-

1

~ I ~ j k ~ i k l

i k l

Other things being equal, the model now tries additionally t o maximize the value of unused resources. The optimal values of x , y are similar t o solutions ( 7 )

and ( 8 )

and we may show that ~ ( h ) is strictly convex as before, with a unique minimum

that now lies in the range

X, >

^- C, for all 1. Should we also wish t o replace the equality resource constraint by the inequality constraint (IS), the appropriate version of the dual constraint (1 7 ) becomes

Note that all three extensions of the simplest model have solutions that are transformations of the simplest solution.

2.4 Solution Procedure

So far we have demonstrated only that all the versions of the model discussed above can be solved by solving equivalent dual problems. In each case we have t o find

X

so as to minimize H(x), sometimes subject t o constraints like (21), but with a unique solution always guaranteed. Because we know the gradient vector and the Hessian matrix

EjhA,

we can begin to search for

^X

by an iterative technique Xi+' = X i

+

t d i (the upper index i denotes the iteration number) which finds better approximations Xi, i = 1, 2,

. . .

, N, t o the solution

X, by taking steps with step-size coefficient t , in the Newton direction

Just two refinements are necessary: first, t o control the step size,and second, t o modify the direction when a constraint like ( 1 7 ) or (21) is appIied and encountered.

In order t o control the step-size coefficient, we need only reduce it if a step seems likely t o overshoot either the solution or a constraint. Figure 2 depicts an appropriate method that tests for this. T o proceed when a constraint like (21) is encountered, we determine the set of resource type indices

where the constraint is active, and where H(X) can decrease only with negative A,. The gradient vector

&

and the Hessian matrix

HAA

are then projected onto the space of active constraints by replacing all the elements corresponding t o active constraints by zeros. They become the reduced gradient vector and Hessian matrix and they determine the Newton direction (22) in the space of inactive constraints

14 i,

which is complemented by zeros for 1 E

L.

Figure 3 shows the complete procedure for determining the optimal X, and hence the solutions i ( X ) , $(XI. A matrix inversion is the only potentially difficult computation. Generally, however, the number of different resource types will be sufficiently small (less than five, say) t o prevent problems.

(Occasionally in the solution of a badly conditioned problem, a step in the Newton direction will not reduce the function H because of numerical errors, and steepest descent d =

- H A

may be necessary.) Note that there is not t o o much extra computation introduced by an inequality resource constraint. Most

Set t = 1

Test for constraint overshoot by checking

+td_ > 0 V I

Test that step n o t too large by checking

slim

H ( h + t d _ ) < H ( h ) + t s Z l - . d , Divide where s is a convergence coefficient (0.3 say)

FIGURE 2 Procedure for determining step-size coefficient t .

of the additional refinements are logical rather than computational. All our applications have been solved by a fairly compact computer program, using no special software; Appendix B gives more details about program size and com- puting efficiency. This program can handle the simple model and all three extensions. In our examples, however, and in most of this report, we refer to the simple model.

3 SOLUTION CHARACTERISTICS

DRAM cannot simulate all patterns of resource allocation that might be observed, and the possibilities for use depend upon the variety of patterns that can be simulated. The analysis given here of admissible solutions t o DRAM is restricted t o the simplest possible DRAM with one patient category, one treatment mode, and one type of resource, for which all the variants described in Section 2 are identical. Section 3.1 shows how the simplest model can be represented graphi- cally, and gives a fundamental condition on admissible solutions. The results indicate the characteristics of solutions for more complex DRAMS, and suggest ways t o fit the model t o small numbers of data points. Sections 3.2 and 3.3 derive conditions for fitting two parameters t o two data points (Appendix C derives conditions for fitting four parameters t o four data points), Section 3.4 derives conditions for fitting two parameters t o one data point, and Section 3.5 derives conditions for fitting four parameters to two data points. These results introduce the next section on parameter estimation from many data points.

3.1 The Simplest DRAM

For the simplest possible DRAM with J = K = L = 1, many elements of the problem can be depicted graphically. First, we can eliminate the Lagrange

I

Choose arbitrary A

I

FIGURE 3 Iterative procedure for solving DRAM.

I 1

Calculate H ( A ) , gradient vector. &

Hessian matrix

1

Determine the set 1 of active resource constraint indices.

Calculate the reduced gradient vector and Hessian matrix

I

Determine the search direction d

1

Determine the step-size coefficient t

N o Take

multiplier between Eqs. (19) and (20), to show how the resource level R (which is input to the model) determines the number of individuals treated x , and the supply level y (which are outputs)

step and

It is easy t o show that these equations have the shapes shown in Figures 4 and 5. Both curves are concave and monotonically increasing.

STOP calculate new A

FIGURE 4 ( x l X ) as a function of ( R I X Y ) .

FIGURE 5 ( y l Y ) as a function of ( R I X Y ) .

Alternatively, we can find an equation that relates x and y directly. The result

where p = ( x / X ) and q = ( y / Y ) is plotted for various ranges of a, in Figure 6.

F o r a

>

- 1, the curve always has just one point of inflection, and when

FIGURE 6 Loci of possible solutions on the x-y plane.

0

^- 1 < a

< P ,

there is just one intersection with the diagonal. From Eq. (25)

whence we deduce that two data points ( p , , q , ) , (p,, q,) can be solutions o f DRAM only if

P z > ~ r - q z > q i (26)

This is a fundamental condition on admissible solutions, which we assume for the rest of this section. It means, for example, that the model cannot repro- duce increasing available hospital beds and decreasing lengths of stay, simul- taneously. (How this condition should be modified when there are two or more resources, perhaps some increasing and others decreasing, is not clear.)

Equation (25) is the locus of solutions of DRAM on the x-y plane. The particular solution for a given resource level is found at the intersection of the locus and the resource hyperbola F(x, y ) = R - xy = 0, and it is the point o n the hyperbola that maximizes the function of Eq. (18). Figure 7 depicts the shape of this function above the x-y plane. We see that

1. U(X, Y) = 0, and x ^-t 0 or y ^-+ 0 implies U(x, y ) ^-t - m. Within the constraints 0

<

^x

<

X, 0

<

y

<

Y, U(x, y ) is always negative and con- cave.

2. U(x, Y) = g(x) and U(X, y ) = h(y). Above the point (X, Y ) the sur- face has gradients

FIGURE 7

0 7 x X

The surface U(x, y ) above the x-y plane.

3 . There is always a unique solution (f,

9 )

because constant-U contours

are always more concave than constant-F curves.

4. Equation ( 2 5 ) is represented by the line OVW.

Evidently, it is not always possible t o choose parameters X, Y, a,

0

that will cause the solution locus OVW t o pass through an arbitrary set of data points ( x i , y i ) , i = 1, 2 , . .

.

, N. In the rest of this section we investigate the conditions that allow this.

3.2 Conditions for Fitting X, Y to Two Data Points

It seems reasonable that at least two points on Figure 6 are needed t o specify a solution locus defined by two parameters, although not all such data will be sufficient or consistent. In this section, we assume that a,

0

are given, together with two data points ( x , , y , ) , ( x , , y , ) such that x , > x , , y ,

>

y , . Can we choose X, Y such that DRAM can reproduce these points? By substituting the two points into Eq. ( 2 5 ) , we easily obtain

The two numerator terms are always positive, and the denominator term is positive if

B <

l n x , - l n x 2 a + 1 In y , -1ny2

This is, therefore, a necessary and sufficient condition for being able t o choose X , Y t o fit two data points.

3.3 Conditions for Fitting a,

P

to Two Data Points

Alternatively, we can assume that X, Y are given, together with two data points (x,, y , ) , (x,, y,) such that x ,

>

x,, y ,

>

y,, and ask whether we can choose a,

such that D R A M can reproduce these points. A necessary and sufficient con- dition for the existence of

0 >

0 is easy t o find. Writing Eq. (25) as

where {(P, q ) = In [ ( l

+

^{(l/P))q-0 -} (l/P)] we can use the two given data points to eliminate a, giving

{(P,q2)-o{(P,41) = O (30)

where o = (In p,)/(ln p , )

>

1. The solution of Eq. (30) is depicted in Figure 8 as the intersection of two curves with known intercepts and asymptotes. There is an intersection for some

0 >

0, if o In (1 - In q , )

>

In (1 ^- In q,) and - o In q ,

<

^- In q,, and these conditions can be combined as

where ^T = (1 - a ) l n q , .

A necessary and sufficient condition for the existence of a

>

0 also comes from Eq. (29). We require that {(P, qi)/(- In p i )

>

1, i = 1, 2. Unfortunately, it is not easy t o remove the dependence o n in this condition. But the two inequalities {(P, q )

>

In (1 - In q ) and {(P, q )

>

^- In q lead to two alternative sufficient (but not necessary) conditions

where we have used the fact that the second condition is stronger for i = 2. We can find a lower bound o n by inspecting the intercepts and asymptotes in Figure 8

In (1 - In q,)

+

omin(- In 9,) = o In (1 - In q , )

0

-

FIGURE 8 Solution of Eq. (30).

When this is used in Eq. ( 3 l) , the two sufficient conditions become

Empirical evidence suggests that the first condition ( 3 2 ) is less restrictive, at least for small values of ( x , lx , ) , and hence closer t o being necessary.

These results suggest the following question. Given four points on Figure 6 , can we align the solution locus through them all? In other words, given four data points ( x i , y , ) , i = 1, 2 , 3 , 4 , with x,

>

x 2

>

x 3

>

x 4 and y ,

>

y 2

>

y 3

>

y 4 , can we choose the four parameters X, Y , a,

p

such that DRAM can repro- duce these points? Sufficient conditions for this, together with an iterative procedure for finding the best fit, are developed in Appendix C . The important conclusion is that even when we have the same number of data points as unknown parameters, and even if the data points satisfy the fundamental con- dition ( 2 6 ) , a perfect fit of the model t o the data is not always possible.

3.4 Conditions for Fitting X , Y or a,

P

to One Data Point

In Section 4 , we use many data points t o estimate pairs of parameters (e.g., X, Y ) by combining the estimates suggested by individual data points. We would expect the conditions for fitting two parameters t o one data point t o be weaker than the conditions derived in Section 3 . 2 for fitting two parameters t o two data points. But is one data point more or less than sufficient t o determine two parameters?

In fact, when a,

0

are given, it is possible t o choose an infinite number of pairs X, Y t o fit a single data point. Equation (25) shows that for any choice of X

>

x , there exists some consistent value of Y

>

y . Similarly, when X, Y are given, it is possible t o choose an infinite number of pairs a, that satisfy Eq.

(25) and that therefore fit a single data point. There is, however, a restriction on the minimum possible values of a,

0:

which can both be zero, only if p ( l - In q ) = 1.

3.5 Conditions for Fitting X , Y , q

0

to Two Data Points

Although we d o not need the result later, it is interesting t o extend and con- clude thls analysis by asking whether all four parameters can be chosen to fit just two data points (xi, y,), i = 1, 2 ; x ,

>

x,; y 1

>

y,. We analyze this prob- lem in two stages. First, can we choose X, Y so as t o satisfy Eq. (3 l ) , the neces- sary condition for the existence of

0 >

O? Second, can we also satisfy Eq. (32) o r Eq. (34), the sufficient conditions for the existence of a

>

O?

In order t o show that we can always choose X, Y consistent with a

0 >

0 , we let w ⁺

=

in Eq. ( 3 1) giving

which can always be satisfied by some T

>

0. In practice, w can be made suf- ficiently large by setting X close t o x , , and the choice of T then determines Y .

In order t o apply a similar procedure t o the sufficient conditions (32) and (34), we write them in the forms

where we have set i = 2 in Eq. (32). Arguing as earlier that we can choose X to make o arbitrarily large, we let w +

=

in these equations

l n y , - - l n y , < ( ~ ) e x p ( r ) - I (37) Combining Eqs. (35), (36), and (37), we have the sufficient condition

Data points in this solution to D R A M

FIGURE 9 Sufficient condition for finding X, Y, a, 0 consistent with two data points (logarithmic scales).

Figure 9 shows the region of (x,/x,, y,/y,) in which a consistent choice of the four parameters X, Y, a ,

0

is always possible.

This analysis shows that two arbitrary data points, even when they satisfy condition (26), may not be consistent with any choice of DRAM parameters X, Y, a ,

0.

It suggests, therefore, that simple procedures that estimate parameters from just two data points (Hughes 1978a) may be unsuccessful. For this reason, we turn to more general methods for parameter estimation.

4 ESTIMATION OF PARAMETERS

We turn t o the problem of calibrating the model, that is, of estimating parame- ters for DRAM appropriate for a given region and policy question. Section 4.1 reviews sources of data. Sections 4.2 and 4.3 then describe separate procedures for estimating the two pairs of parameters X, Y and a ,

0,

which are drawn together in Section 4.4. These procedures are quite suitable for small examples and they are illustrated in Section 5. Section 4.5 outlines an alternative approach t o parameter estimation that incorporates specific assumptions about the uncertainty of model predictions. It shows that, with certain approximations, the approach is feasible and worth testing. Section 4.6 concludes by briefly mentioning the problems of estimating resource costs.

4.1 Parameters and Data

The parameters of the model fall into three groups:

a The ideal levels X, Y at which patients would be admitted and receive resources, if there were no constraints on resource availability. Abso- lute values of these parameters have little meaning, but relative values can be chosen to indicate the relative "needs" for health care.

The power parameters a , which reflect the elasticities of the actual levels to changes in resource supply. For example, we expect the elas- ticity of admission rate t o bed availability to be less for appendicitis patients than for bronchitis patients, because appendicitis usually requires faster attention.

The relative costs C of different resources. DRAM uses the marginal unit cost of a bed-day, a doctor-hour, and so on, or equivalent parame- ters, in order to choose between alternative mixes of these resources.

We defer discussion of resource cost estimation until Section 4.6.

The level of available resources is not regarded as a model parameter but as an experimental variable. DRAM shows how the levels of satisfied demand vary with changes in resource supply.

There are more data available t o estimate X, Y, ^cu,

P

than there are for many other problems in HCS modeling. The sources include:

Other models a Special surveys

Professional opinions a Routine statistics

At IIASA, other models have been developed for other components of the HCS, and particularly for the estimation of true morbidity from degenerative diseases (Kaihara et al. 1977) and infectious diseases (Fujimasa et al. 1978). Later at IIASA, these outputs may be useful for setting the ideal rates at which patients in different categories need care. Initially, however, we wish to test and use DRAM independently of other models. Many researchers have performed important and useful special surveys. Among others, Newhouse and Phelps (1974) and Feldstein (1967) have estimated both elasticities in hospital care and the costs of acute services, and some of these results were used t o calibrate a Mark 1 version of DRAM (Gibbs 1978b). Unfortunately, these results may not be relevant in other regions or countries, or at other times. In an international setting it is necessary to avoid relying on results related t o a specific health system.

The professional opinions of doctors and health planners can be useful for setting ideal levels of care. Countries where there is a high degree of central planning often set normative figures for ideal hospitalization rates and neces- sary standards of care, and these can be used in DRAM. However, these are not available in all countries, and probably no professional should be asked t o esti- mate elasticities, in case he supplies his own rather than those of the HCS. This

leaves routine statistics. Most systems keep regular records o n the use and costs o f their services, and o n how they have allocated resources in the past. If DRAM is a valid model of the HCS, then these figures are typical outputs of the model, which we should be able t o use for model calibration.

The aim of DRAM is t o model how the HCS reacts t o change. Generally, therefore, DRAM'S model parameters must be estimated from data that show how an unchanging HCS reacts t o external changes, either in space or time.

Cross-sectional data from subregions of the region of interest may show the HCS operating at different resource levels. So also may longitudinal data col- lected at different times. In both cases, however, the underlying system may be different for the different data. Subregions are often deliberately defined so as t o be predominately urban or predominately rural, and we must consider ways of averaging the results across the region. Data collected at different times are highly likely t o be affected by historic trends in medicine o r management.

Ideally, we should model these trends and incorporate the time-varying parame- ters in a time-dependent model. More probably, we shall use data from a period during which we can assume time variations t o be small. The resulting model will still be good for representing those aspects of resource allocation behavior that are independent of time trends. A final and obvious problem is that the available data may be incomplete, either because of recording failures o r because the data is insufficiently disaggregated.

Not all of these problems can be overcome simultaneously. But in the next three sections we concentrate o n estimation methods that are based on routine statistics about current o r past allocation behavior, and that take into account that cross-sectional and longitudinal data may reflect inherent param- eter variations. In addition, one of the procedures can be used with incomplete data.

4.2 Estimation of X , Y

We consider first the estimation of the ideal service levels X and the ideal supply levels Y, assuming for the moment that the power parameters a, /3 are known.

Sufficient information t o estimate X , Y is given by the current allocation of resources in the region under study. If the current allocation pattern is described by x and y , Eqs. (7) and (8) may be rearranged as

which are expressions for X and Y. We have a single equation for each unknown parameter, but as Section 3.4 predicted, we still need some external criterion t o determine h. If we assume that we can define the resources needed t o satisfy the ideal levels Xjk, Yjkl as some multiple 8 , of t h e resources used currently

then (39) and (40) can be substituted into (41) t o give

where fl(h) = 0 V l

and where Eq. (42) must be solved for A. T h e equatioils in f are very similar t o the equation H, = 0 that arises during model solution, and, provided that 8,

>

1, V I , and that all the terms except h are known, they may be solved in the same way t o give h . Unfortunately, not all the terms are known. In partic- ular, pjk is a weighted average iilvolving the terms Yjkl, which are as yet unknown.

It is therefore necessary t o iterate between solving Eq. (42) for A, and Eqs. (39) and (40) for X , Y.

This approach suffers from the disadvantage that it only finds values of X , Y that are consistent with t h e current allocation pattern and the assumed values for a,

0.

A model with parameters estimated o n so little data may have little predictive power. More useful is t o estimate X , Y from o t h e r data and then t o use the current allocation as a test of t h e model's validity. Other suit- able data include cross-sectional and longitudinal data, and given

N

data points from such sources, we can use Eqs. (39) and (40) t o find N estimates of X, Y.

T h e problem remains of how t o combine these estimates.

Estimates Xjk(i), Yjkl(i) derived for subregions i = 1,

. . .

,A', may be combined rather easily. If the population of subregion i is P(i), then Xjk(i)P(i) is t h e number of individuals in category j in mode of care k who need treat- ment in subregion i (per year), and Xjk(i)Yjkl(i)P(i) is the number of resources I needed t o treat these individuals (per year). These quantities may be summed across the region, and t h e corresponding regional estimates of X , Y are

This approach (also depicted in Figure 10) is interesting because we d o not need t o assume that X , Y are constant across the region. The subregional variations are averaged by summing the ideal demands across the region.

Estimates Xjk(i), Yjkl(i) derived at different times i = 1, .

.

. , N are more difficult t o combine. Ideal supply levels Yjkl are probably decreasing with time, and an exponential curve could be fitted t o a long sequence of points. T h e ideal numbers of patients needing care per head of population, Z, = Z k Xjk, V j , will change because of changes in the age structure and in the morbidity rates. We call correct for t h e former, but the latter are affected by changes in doctors'

knowledge of a, l3.C

arbitrary con- straint upon h

for each data

Population Combine

for each to find

average X.Y

polnt

FIGURE 10 Estimation of ideal levels.

preferences between modes of health care. These are reflected in the values of Xjk, which could, if necessary, be regarded as experimental variables.

4.3 Estimation of a,

0

We now consider how t o estimate the power parameters a ,

0,

assuming for the moment that the ideal levels X, Y are known. Sufficient information t o estimate a, /3 is given by the current allocation of resources in the region under study. If the current allocation pattern is described by x and y, Eqs. (7) and (8) may be

which are expressions for a and

0.

As in Section 4.2, A must be determined externally. We know, however, that a and

p

are always positive. This implies

and we can conveniently define A, as some (small) multiple 4 ,

>

1 of the mini- mum value

X I

A, = $,jll V l (46)

A second problem is that Eq. (44) gives K values for each aj. Generally, these will be different values, but we can overcome t h s by aggregating the data across modes, and by using Eqs. (44) and (45) for one super mode.

By these means, we may estimate values for the parameters a,

0.

The

model so calibrated will not exactly reproduce the current allocation of resources unless the latter is one of the admissible solutions of DRAM defined in Section 3 . However, it will reproduce the actual supply levels y j k l , and the actual numbers of patients in each category (xi,

+

xj2

+ . . . +

xi,). Whether the esti- mated elasticities are useful for forward prediction will depend upon whether the current allocation pattern is representative of the HCS's usual behavior. The procedure described above only finds values for a,

0

that are consistent with this assumption and with the values assumed for X, Y.

A more sophisticated approach is to use more data by estimating empirical elasticities. These can then be used to derive the power parameters a,

0.

Appro- priate empirical elasticities for DRAM are yjkl, the elasticity of the service level xjk t o changes in the resource level R , , and q j k m l , the elasticity of the supply level y j k , t o changes in the resource level R , . They can be predicted for given resource levels by DRAM. For example, yjk, is defined as

We use Eq. (7) t o get an expression for

a

In x j k / a p j k . Thus,

Similarly, where

- R ,

a x ,

qjkrnl =

( P j k l +

aR

¹

and the derivatives aR,/aX, = a2H/aXlaX, are given by Eq. (13). Equations (47) and (48) can be written as

Ajkl

a. = - - - - I (49)

yjkl

where

and where

I,,

is element ml of the inverted Hessian matrix. However, solution for a,

0

is still hard. First, this is because A and B are functions of a and

0 ,

and iterative solution is necessary. Second, X must still be chosen externally, and the empirical elasticities must be consistent with the choice of A, otherwise the procedure may not converge (Gibbs 1978b). Third, there are more empirical

elasticities y, q than there are power parameters a , 0. Therefore, unless some of the empirical elasticities are ignored, the parameters will be overspecified.

Fourth, the empirical elasticities y, q , are not directly measurable and are usually the result of some prior data analysis.

Some of these difficulties can be avoided by incorporating the prior data analysis within the solution of Eqs. (49)-(52). For example, estimates y, q are found by assuming that some

N

known data points xjk(i), yjkl(i), RIG), i = 1,

.

.

. ,

N ,

satisfy the linear models

in which a x , aY are unknown constants, and e x , eY are random, uncorrelated error terms with zero means. If we eliminate y, q by combining Eqs. (49), (50), (53), and (54) t o give

we can use the following iterative scheme in order to estimate a and 0.

1. Fix h arbitrarily for some resource level R , perhaps by using Eq. (46) on one of the data points.

2. Assume some initial estimates of a ,

0

(e.g., unity).

3. Derive p from Eqs. (9) and (1 O),

Axx

from Eq. (1 3), and A , B from Eqs. (5 1) and (52).

4 . Find the best least-squares estimators of (aj

+

1)-', (Pjkm

+

I)-' in Eqs. (55) and (56).

5. Hence, estimate a ,

0

and repeat from step 3.

This procedure (also depicted in Figure 11) is likely t o be lengthy because it incorporates regression estimation at each iteration. Nor can we ensure the positive estimates of a ,

0

that are necessary for convergence. On the other hand, it has the advantage that more of the original data can be used directly. If a full data set

{xjk(i),yjkl(i)R1(i); i = I, . . . ,

N,j

= 1 , , . . , J

is available, KN equations are available t o estimate each a,, and perhaps not all of the xjk(i) need be known. Fewer equations oust N) are available t o estimate each Pjkl, and it may be necessary t o introduce some further simplifying

a t some known resource level

Calculate

A

(P + I I-' via regression

FIGURE 11 Estimation of power parameters.

assumptions such as = fljXkl, V j l , j l E (1,

. . .

, J ) , in order t o obtain reliable estimates. A second advantage of this procedure is that it is not necessary t o modify any of the input data t o make them consistent with the model. A third advantage is that the parameter estimated in each regression has an estimated standard error associated with it. These errors provide a measure of the reliabil- ity of a,

0.

Perhaps the main assumption in the above analysis is that the underlying elasticities are constant across the set of data points. Because there is little information about how elasticities are likely t o vary in time or space, we have not attempted t o model this variation here. But Appendix D shows that in a certain sense, the procedure described above gives unbiased estimates. This is a reassuring result, and the estimates can be further tested t o see if the model so calibrated can reproduce the current allocation of resources.

4.4 Estimation o f a,

0

and X , Y

In the most general case, neither of the parameter pairs X, Y or a ,

0

is known, and we require estimates of both. In this circumstance, the two procedures described above may be used together in the following scheme.

1. With some arbitrary initial estimates of ^a,

0,

use the methods of Section 4.2 t o estimate X, Y.

2. With these estimates of X, Y, use the methods of Section 4.3 to esti- mate a ,

p.

3 . Repeat from step 1.

The analysis in Section 3 showed that not even all small data sets can be con- sistent with DRAM, so that convergence of this scheme cannot be guaranteed.

For this reason, although we have implemented on the computer the procedures for estimating both X, Y and a,

0,

we prefer not t o link these programs together, but rather t o use them alternately t o obtain consistent pairs of estimates. (Note however that when neither parameter pair is given exogenously, the same data cannot be used t o estimate both pairs of parameters.)

The parameter estimation procedures described above involve the choice of additional constraint variables such as 4 and 19. Fortunately, however, this is not a problem. Although different values of 4, I9 lead t o different values for a,

p,

X, Y, each set of parameter values will reproduce with similar accuracy the data points used for estimation. Provided that predictive runs of the model do not involve resource levels very different from those used in estimation, the results will be relatively insensitive t o 4, 8. Section 5 illustrates how these pro- cedures were used t o estimate model parameters in two examples.

4.5 An Alternative Approach

We now describe an alternative approach t o parameter estimation that takes into account that DRAM'S predictions are subject to uncertainty, and that incorpor- ates this uncertainty mathematically. It is not fully implemented or tested, but the preliminary analysis given below is encouraging.

We consider how t o use historical resource allocations x ( i ) , y ( i ) , i = 1,

. . .

, N in order t o estimate the model parameter set P = {X, Y, a,

p } .

As mentioned in Section 4.1, these are not the only data available. Nor does P include all the parameters: we have omitted the resource costs C because they seem to be more naturally estimated from external studies of financial or related statistics. Nevertheless, procedures t o estimate these parameters from these data would be useful.

If reality conformed exactly to DRAM, we would expect the historical allocations x(i), y(i) t o be exactly those i ( i ) , $(i) prescribed by DRAM for the historical resource levels. These solutions are the result of (constrained) maxi- mization over x and y of a function U(x, y , P, C, R ) that depends also upon the parameters P, the costs C, and the resource levels R . This function is known, and is presumably also maximized by choosing the correct parameters

max , U ( x , y , P , R , C )

pgiven past { ~ . ~ ' , R , C J

because with wrong parameters, it would be maximized by different values of X,Y.

However, DRAM is only a model of reality. The historical allocations are related t o the model predictions by equations like x ( i ) = f ( i )

+

t , ( i ) and y ( i ) = y ( i )

+

t 2 ( i ) where t l ( i ) , t2 ( i ) are stochastic processes with statistics S

that need t o be specified. Such a specification would be quite complicated. The probability distributions involved in S depend upon the reasons why the assumptions in DRAM are not perfect, the reasons that influence actual decisions, and the reasons that give rise t o inaccurate data. But if such a specification were possible, the parameter set P could be estimated through

max conditional U ( f , j j , P , R , C )

P expectation ( 5 7 )

with respect t o i . y given x, Y , S

where

U ( f , p , P , R , C ) = max U ( x , y , P , R , C ) ( 5 8 )

x , y iven past

f ~ ,

^CI

Such a calculation would also be quite complicated, however, because the integral involved in the conditional expectation is unlikely to be analytic. In short, the ideal estimation procedure is extremely difficult both t o formulate and solve. It does, however, suggest a more practical approach.

If the function U ( f , j j , P , R , C ) in Eq. ( 5 8 ) were twice differentiable in f , j j , it could be expanded as a Taylor series about the point x , y , with terms

in the prediction errors (i - x ) , - y ) . If, in addition, S were such that EXPECTATION t l ( i ) = EXPECTATION t 2 ( i ) = 0 , term-by-term expansion of this series would eliminate all first-order terms, causing the dominant terms of the series to be the squares and cross-products of the prediction errors. Whereas this is hardly a feasible way t o solve ( 5 7 ) , it suggests the idea of formulating the parameter estimation problem as the minimization of a function of the squared prediction errors

m in J(P)

P ( 5 9 )

where

1 1

J ( P ) = -

C

p$k[fjk(i) - xjk(i)12

+

^-

2

^{~ $ k j} [ j j k j ( i ) -yjkl(i)12 ( 6 0 )

2 ^ijk 2 ^ijkl

in which

f ( i ) , j ( i ) are the optimal model allocations for assumed P and known past R ( i ) , C ( i ) , i = 1,

. . .

, N

x ( i ) , y ( i ) are the observed historical resource allocations for known past R ( i ) , i = 1 , . . . , N

p$,, p h , are weighting coefficients t o be specified later.

DRAMS most useful feature is that the solutions f , j j are analytic functions of the parameters P. This means that we can calculate the gradient vector and Hessian matrix of J ( P ) , opening the way for powerful techniques for solving ( 5 9 ) . The gradient vector is

and the Hessian matrix is

a2Pjkl(i) a j j k l ( i ) aPjkl(i)

j j

+ )

^{( 6 2 )}

ijkl apt a p

if the prediction errors are small. Expressions for the elements in the sensitivity derivative vectors aiik(i)/aP and a j j k l ( i ) / a P are evaluated and listed in Appendix E.

The dimension of these vectors, and also of the Hessian matrix, is the same as the number of parameters ( 2 J K L

+

JK

+

J ) in the parameter set P. Each element in the Hessian matrix is the sum of the N(JK

+

J K L ) terms enumerated in Eq. ( 6 3 ) . Renumbering these terms as m = 1 , 2 ,

. . .

, N(JK

+

J K L ) , we obtain the simpler form

a

^J(P)

- -

-

C

^prnvrnvk

aPtaP where p are scalars

P I = p;11, ^{P 2} = p;12,

. . .

By arguments similar t o those in Section 2 . 2 , a matrix such as Eq. ( 6 4 ) is always positive semidefinite, which is useful for search procedures to solve (59).

However, the Hessian matrix will not be positive definite, and such searches will fail, unless the vectors urn are linearly independent and span the parameter space. Just 2 JKL

+

^JK

+

J parameters X j k , Y j k l , ctj,

Ojkl,

V j , k, _I, have t o be esti- mated, and each data point x j k , y j k l , v j , k. I. provides JK

+

JKL degrees of free- dom that are subject t o L resource constraints. Therefore, the number of data points N needed t o identify P m u s t satisfy N(JK

+

^{JKL - L ) 2 2JKL}

+

^JK

+

J . When J = K = L = 1, N must be f o u r o r m o r e , but w h e n J = K = 3 a n d L = 2 , N can be as small as 2 , although more data than this would be needed t o achieve reasonable confidence in the estimated parameters.

An attempt t o choose parameters P that will minimize J(P) may also fail if the problem is badly conditioned, and specifically if the eigenvalues of